For Bernouli map
\(f(x)=r x\vert _{mod 1}, r>0\) , we can guess that for \(r<1\) , the fixed point is stable
for any value of \(\alpha \) .
This is precisely what Deshpande and Gejji [1]} find.
In the
case of Gauss map given by
\(f(x)=\exp (-7.5 x^2)+\beta \) , the fixed point can be found numerically
using bisection method or other methods. The fixed point is close
to \(\beta \) . The slope is negative and the stability regime is
dependent on value of \(\alpha \) . One expects this fixed point to
be more stable if \(\alpha \)  increases. It can be
checked that \(a_c(0.4)=-0.3132\) , \(a_c(0.6)=-0.5085\) 
and \(a_c(0.8)=-0.7328\) . The critical points
corespond to values of \(\beta \)  at which the \(f^{\prime }(x^*)\)  matches with
these values. Thus we expect the fixed point to
be stable for \(\beta >0.49\)  for \(\alpha =0.8\) , for \(\beta >0.57\)  for \(\alpha =0.6\) 
and \(\beta >0.65\)  for \(\alpha =0.4\)  (cf. Fig. REF ).
